Multiple singularities of the equilibrium free energy in a one-dimensional model of soft rods

There is a misconception, widely shared amongst physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at non-zero temperatures, can not show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counter-example. We consider thin rigid linear rods of equal length $2 ell$ whose centers lie on a one-dimensional lattice, of lattice spacing $a$. The interaction between rods is a soft-core interaction, having a finite energy $U$ per overlap of rods. We show that the equilibrium free energy per rod $mathcal{F}(tfrac{ell}{a}, beta)$, at inverse temperature $beta$, has an infinite number of singularities, as a function of $tfrac{ell}{a}$.